- As Phil indicated, it doesn't take much time or brain merely

to *find* a PrP of the type that I guess Harvey has in mind

to prove. It was as simple as ABC to find

(1579393975)_{1386}1

which is a 13861-digit palindromic PrP,

all of whose digits are odd,

with no neighbouring digits equal

(contrast with top-20 palindromes)

PFGW Version 20010818.Win_Dev

(Beta software, 'caveat utilitor')

Primality testing 15793939750*R(13860)/R(10)+1

[N-1, Brillhart-Lehmer-Selfridge]

Reading factors from helper file HD13860.fac

Running N-1 test using base 3

Running N-1 test using base 7

Running N-1 test using base 11

Calling Brillhart-Lehmer-Selfridge with factored part 29.59%

15793939750*R(13860)/R(10)+1 is PRP!

(2206.190000 [Rosinante] seconds)

It may be proven prime with 4 more things:

another prime factor of 10^13860-1;

and another (if first less than p57);

a rerun of Pfgw with these factors;

a Konyagin-Pomerance cubic test.

David - I got trusty old Rosinante to redo all the Pari/Pfgw/Tx

tests for the 13861-digit all-odd-digit palindrome

(1579393975)_{1386}1 = 15793939750*(10^13860-1)/(10^10-1)+1

and then I packed them in

http://groups.yahoo.com/group/primenumbers/files/Factors/djb13860.zip

together with a demo that my Pari KP code replicates

Andy Steward's largest-ever KP test (at 10619 digits).

The KP cubic for the palindrome disagrees with Satoshi's.

But maybe he did not use all the factors?

Satoshi: please check that you ran with

F1=log(F)/log(N)=0.3012668 where F contains

all known prime divisors of N-1.

Did you include repeats,

and also factors of 15793939750?

I did.

David - David, be glad!

I tested using file hd13860p.fac including factors of 15793939750.

The result by my program conincides with PARI's output at any point.

One of the roots is 3.309279218*10^(-1421).

> Did you include repeats,

Yes, previous test didn't use factors of 15793939750.

> and also factors of 15793939750?

> I did

But it was a 30.05% proof.

Satoshi Tomabechi